Abstract:We implemented an algorithm that allow us to find error correcting codes for which the minimum distance is bigger possible in a fixed alphabet <img src= image/apps/mmdc216.png > , with fixed length n an dimension k. <br>
The alphabet is the finite field with q elements. We proceeded in several steps:<br><br>
1. We construct all the sets with k elements of the n dimensional vector space over the alphabet;<br>
2. We determine which of these sets are linearly independent. Of course, they span k dimensional subspaces (even if someone of them can be equal);<br>
3. We get all the <img src= image/apps/mmdc16.png > (Gauss coefficient) distinct subspaces; They are linear codes of length n and dimension k;<br>
4. In the set of these subspaces, we choose the ones with minimum distance higher. <br><br>
Note that steps 1, 2 and 3 allows us to construct, in an exhaustive way, all the subspaces with fixed dimension. Hence, there is no way to determine linear codes with same length and dimension but with higher minimum distance compared with the one we found with our algorithm.
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We implemented a serial version of the algorithm and we tried it successfully in a personal computer with a single processor. We want to implement the parallel version.
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<b>Software requirements</b>: Parallel MATHLAB, C, and MPI^